I finally was able to figure out how to assign a permission to a group and allow them to see the kit. I was confused at first because the server I saw this on originally had all the kits visible to all default users but did not allow them to use the kit. How would that be configured so that it is visible for everyone but not allow them to use unless they are put into the proper permissions group?

Side note... I am still confused about the difference between Authlevel and Permissions. What does the Authelevel essentially do and how would someone get that specific authlevel?

Authlevels is either 0 which is everyone 1 which is mods or 2 admins

Permissions rely on oxide which can have a lot more groups.

The solution is simple , you make the kit hidden false when you edit it.

/kit edit <kitname>          a menu will appear
/Kit Hidden false.              a confirmation will appear

As for permissions you create the permission such as "highendkit" and you give a group that permission
/o.grant group <groupname> kits.highendkit               a confirmation will appear that the group has that permission.
make sure you create the group first !

I was thinking that the hide function would make it visible. But that doesnt seem to be the case. When I do /kit hide false it successfully makes the change. But people without the permission are unable to see it in the menu. As soon as I add them to the permission group they are able to see the options.

check the config you can enable the view for people that don't have permissions